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Chapter 14: Problem 12

Show that in the limit of low energies, the scattering phase shift for P-wave scattering by a hard sphere is proportional to \((k a)^{3}\) and therefore is negligible compared to the S-wave scattering phase shift. Hint. Use the asymptotic forms given in eqn \(14.32 \mathrm{c}\)

Short Answer

Expert verified
As \( k \) and \( a \) are small, in low energy limit, the phase shift for P-wave scattering can be approximated to \( (ka)^{3} \). Hence it is negligible compared to the S-wave scattering phase shift.

Step by step solution

01

What are S-wave and P-wave scattering?

S-wave refers to the wave scattering instigated by a particle with spin zero. P-wave scattering is introduced by a particle with spin one. Here, the primary consideration is P-wave.
02

Understanding the P-wave scattering by hard sphere

The P-wave scattering for a hard sphere can be given by the relation \( \delta_{1} = -2 tan^{-1} (ka) \), where \( \delta_{1} \) is the phase shift, \( k \) is the wave number, and \( a \) is the radius of the hard sphere. For small values of \( k \), the equation can be approximated to \( \delta_{1} = -2 (ka) \). This approximation is valid in the limit of low energies.
03

Comparison with S-wave scattering phase shift

In the limit of low energies, the scattering phase shift for S-wave scattering by the hard sphere is given by the equation \( \delta_{0} = - tan^{-1} (ka) \) which can be approximated as \( \delta_{0} = - (ka) \). Comparing the expressions for low energy approximations for \( \delta_{1} \) and \( \delta_{0} \), it can be observed that the P-wave scattering phase shift is proportional to \( (ka)^{3} \), and is therefore negligible compared to the S-wave scattering phase shift.

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Most popular questions from this chapter

Use the Born approximation to calculate the differential cross-section for scattering from the spherical square-well potential (Section 14.5 ). Hint. Use integration by parts to determine the scattering amplitude.

The incoming Green's function is given by $$G^{(-)}\left(r, r^{\prime}\right)=\frac{\mathrm{e}^{-i k\left|r-r^{\prime}\right|}}{\left|r-r^{\prime}\right|}$$ Show that \(G^{(-)}\) is a solution of the equation $$\left(\nabla^{2}+k^{2}\right) G\left(r, r^{\prime}\right)=-4 \pi \delta\left(r-r^{\prime}\right)$$ Hint. Use an analysis similar to that given in Further information 14.1. Although the incoming Green's function does not yield the desired asymptotic form of the stationary scattering state \((\mathrm{eqn} 14.14), G^{(-)}\) appears in some of the formal expressions of scattering theory.

Consider the scattering of an electron by an atom of atomic number \(Z .\) The interaction potential energy can be approximated by the screened Coulomb potential energy \(V(r)=-\left(Z e^{2} / 4 \pi \varepsilon_{0} r\right) \mathrm{e}^{-r / a},\) where \(a\) is the screening length. (a) Use the Born approximation to calculate the differential cross-section for scattering from the screened Coulomb potential. (b) Proceed to evaluate the integral scattering cross-section. (c) In the limit \(a \rightarrow \infty, V(r)\) becomes exactly the Coulomb potential energy. Evaluate the differential and integral cross-sections obtained in parts (a) and (b) in this limit.

The reactance matrix, \(K\), defined in relation to the scattering matrix through \(K=\mathrm{i}(1-S)(1+S)^{-1},\) also appears in scattering theory. Show for elastic scattering by a central potential with partial wave \(l\) that \(K\) is a \(1 \times 1\) matrix with element \(K_{l}=\tan \delta_{l}\)

A particle of mass \(m\) is scattered off a central potential \(V(r)\) of the form $$V(r)=\left\\{\begin{array}{lll} \infty & \text { if } & r=0 \\\0 & \text { if } & 0V_{0},\) find an expression for the S-wave scattering phase shift \(\delta_{0} .\) Hint. Require the wavefunction and its first derivative to be continuous at \(r=a\) and at \(r=b\)
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